Episode 8 — Aptitude and Reasoning / 8.2 — Profit and Loss

8.2 Quick Revision -- Profit and Loss

One-page formula sheet for last-minute revision before exams.

1. Core Definitions

CP  = Cost Price (price at which goods are bought)
SP  = Selling Price (price at which goods are sold)
MP  = Marked Price / List Price (price on the label)

Profit = SP - CP       (when SP > CP)
Loss   = CP - SP       (when CP > SP)

2. Relationship Diagram

                  + Markup %                 - Discount %
          CP  ─────────────────>  MP  ─────────────────>  SP
          ^                                                |
          |                                                |
          +─────────── Profit (+) or Loss (-) ─────────────+
                      (always on CP base)

KEY RULES:
  * Profit / Loss %  -->  always calculated on CP
  * Discount %       -->  always calculated on MP
  * Markup %         -->  always calculated on CP

3. All Formulas at a Glance

Profit and Loss

Profit %  = ((SP - CP) / CP) x 100
Loss %    = ((CP - SP) / CP) x 100

Finding SP

SP = CP x (100 + Profit%) / 100       ... (profit)
SP = CP x (100 - Loss%) / 100         ... (loss)

Finding CP

CP = SP x 100 / (100 + Profit%)       ... (profit)
CP = SP x 100 / (100 - Loss%)         ... (loss)

Markup and Discount

MP = CP x (100 + Markup%) / 100
SP = MP x (100 - Discount%) / 100

Markup %   = ((MP - CP) / CP) x 100
Discount % = ((MP - SP) / MP) x 100

Combined Markup-Discount (Net Effect)

Net % = M - D - (M x D) / 100

    Positive result = Profit %
    Negative result = Loss %

Successive Discounts

Two discounts d1%, d2%:
    Equivalent single discount = d1 + d2 - (d1 x d2) / 100

Dishonest Dealer

Profit % = (Error / False Weight) x 100
         = ((True Wt - False Wt) / False Wt) x 100

With claimed profit/loss of P%:
    Multiplier = (True Wt / False Wt) x (100 +/- P) / 100
    Actual %   = (Multiplier - 1) x 100

Two Articles, Same SP, Equal x% Profit and Loss

Net result = ALWAYS a LOSS

Loss % = x^2 / 100

Buy X Get Y Free

Effective Discount = (Y / (X + Y)) x 100

Buying and Selling at Different Rates

Buys 'a' for Rs. P, Sells 'b' for Rs. Q:
    Profit/Loss % = ((BX - AY) / AY) x 100
    
    where B = Q, A = P, X = a, Y = b

Same amount shortcut (Rs. X buys 'a', Rs. X sells 'b'):
    a > b:  Profit % = ((a - b) / b) x 100
    a < b:  Loss %   = ((b - a) / b) x 100

Partnership

Profit Ratio = (Capital_1 x Time_1) : (Capital_2 x Time_2) : ...

Profit/Loss as Fraction of SP

Profit = (1/n) of SP  -->  Profit % = 100 / (n - 1)
Loss   = (1/n) of SP  -->  Loss %   = 100 / (n + 1)

Conditional SP Problems

If selling at a% gives SP1 and selling at b% gives SP2:
    CP = (SP1 - SP2) x 100 / (a - b)        [use signed values for a, b]

If a% profit and a% loss:
    CP = (SP_profit + SP_loss) / 2

4. Fraction-Percentage Shortcut Table

 %       Fraction    Profit Multiplier    Loss Multiplier
------   --------    -----------------    ---------------
  5%      1/20            21/20               19/20
 10%      1/10            11/10                9/10
 12.5%    1/8              9/8                 7/8
 16.67%   1/6              7/6                 5/6
 20%      1/5              6/5                 4/5
 25%      1/4              5/4                 3/4
 33.33%   1/3              4/3                 2/3
 40%      2/5              7/5                 3/5
 50%      1/2              3/2                 1/2

5. Common Dishonest Weight Table

 False Weight (per kg)  |  Profit %
 -----------------------|-----------
       950g             |   5.26%
       900g             |  11.11%
       850g             |  17.65%
       800g             |  25.00%
       750g             |  33.33%

6. Combined Markup-Discount Quick Reference

Markup %  | Discount %  | Net Effect
----------|-------------|------------
  20%     |   10%       |  +8% profit
  25%     |   20%       |   0% (break-even)
  30%     |   20%       |  +4% profit
  40%     |   20%       |  +12% profit
  50%     |   30%       |  +5% profit
  40%     |   25%       |  +5% profit
  50%     |   20%       |  +20% profit

7. Buy X Get Y Free -- Quick Values

Offer              | Effective Discount
-------------------|-------------------
Buy 1, Get 1 Free  |  50%
Buy 2, Get 1 Free  |  33.33%
Buy 3, Get 1 Free  |  25%
Buy 4, Get 1 Free  |  20%
Buy 5, Get 1 Free  |  16.67%
Buy 9, Get 1 Free  |  10%

8. Same SP Profit-Loss Table

Equal Profit & Loss %  |  Net Loss %
-----------------------|-------------
      10%              |    1%
      15%              |    2.25%
      20%              |    4%
      25%              |    6.25%
      30%              |    9%
      50%              |   25%

9. Common Exam Patterns -- Quick Recognition

Pattern	Formula
Markup + Discount	Net = M - D - (MD/100)
Dishonest dealer at "cost price"	Profit% = Error / False Wt x 100
Same SP, equal profit/loss	Loss% = x^2/100
Buy X Get Y Free	Discount = Y/(X+Y) x 100
SP increased by Rs. N, profit changes by P%	CP = N x 100 / P
Sells fraction at profit, rest at loss	Weighted average of percentages
Chain of dealers each adding profit	Multiply all (1 + p/100) factors
Break-even markup for given discount	M = 100D / (100 - D)
Break-even discount for given markup	D = 100M / (100 + M)

10. Critical Reminders

1. Profit/Loss % is ALWAYS on CP (unless explicitly stated otherwise).
2. Discount % is ALWAYS on MP.
3. Successive discounts of a% and b% != single discount of (a+b)%.
4. Same SP with equal profit% and loss% ALWAYS gives net LOSS.
5. When in doubt, let CP = Rs. 100 and work with percentages directly.
6. "Sold at cost price" does NOT mean no profit -- check for false weights.
7. Overhead expenses ADD to cost price.
8. In partnership, profit ratio = (Capital x Time) ratio.